Average Error: 15.1 → 0.0
Time: 1.2s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)
double f(double x, double y) {
        double r548994 = x;
        double r548995 = y;
        double r548996 = r548994 + r548995;
        double r548997 = 2.0;
        double r548998 = r548994 * r548997;
        double r548999 = r548998 * r548995;
        double r549000 = r548996 / r548999;
        return r549000;
}
↓
double f(double x, double y) {
        double r549001 = 0.5;
        double r549002 = 1.0;
        double r549003 = y;
        double r549004 = r549002 / r549003;
        double r549005 = x;
        double r549006 = r549002 / r549005;
        double r549007 = r549001 * r549006;
        double r549008 = fma(r549001, r549004, r549007);
        return r549008;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.1
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.1

    \[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2) y)))